Assessment and optimization for metrology instrument including uncertainty of total measurement uncertainty

ABSTRACT

Methods and related program product for assessing and optimizing metrology instruments by determining a total measurement uncertainty (TMU) based on precision and accuracy. The TMU is calculated based on a linear regression analysis and removing a reference measuring system uncertainty (U RMS ) from a net residual error. The TMU provides an objective and more accurate representation of whether a measurement system under test has an ability to sense true product variation. The invention also includes a method for determining an uncertainty of the TMU.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of Ser. No. ______ AttorneyDocket No. BUR920020118US1, filed Feb. 10, 2005, which claims benefit ofP.C.T. Application No. PCT/US2002/041180, filed Dec. 20, 2002. Thisapplication also claims benefit of U.S. Provisional Patent ApplicationNos. 60/546,590 and 60/546,591, both filed Feb. 20, 2004, each of whichis hereby incorporated herein.

BACKGROUND OF THE INVENTION

1. Technical Field

The present invention relates generally to metrology instruments.

2. Background Art

Efficient semiconductor manufacturing requires highly precise andaccurate metrology instruments. In particular, a metrology instrument isrequired to achieve small tolerances to achieve better quality productsand fewer rejections in the manufacturing process. For example, the 1999Edition of the International Technology Roadmap for Semiconductors liststhe necessary precision needed for isolated line control in the year2001 to be 1.8 nm. Unfortunately, correctly assessing and optimizing themeasurement potential of a metrology instrument is difficult for anumber of reasons. For example, an evaluator normally has limited accessto the various instruments under consideration. In addition, eachinstrument needs to be evaluated under a wide range of conditions inorder to gain a valid impression of how it will perform in the actualmanufacturing setting. Finally, there are no widely accepted standardsrelative to the required parameters and how the parameters should bemeasured. As a result, an adequate solution for calculating anuncertainty of a metrology instrument in meaningful units of length forcomparison to manufacturing lithography requirements has been elusive.

Current assessment methods are often based on the repeatability andreproducibility (R&R) of an instrument. For a critical dimension (CD)metrology instrument, evaluation is often executed by pullingrepresentative samples of partially constructed product wafers from amanufacturing line. Recipes (programming instructions) are thenimplemented on an instrument under evaluation such that estimates of thestatic repeatability and long term reproducibility can be made. Forexample, to determine static repeatability for a measurement of a givenproduct level, a recipe is implemented to cause the CD metrologyinstrument to navigate to a particular site on the wafer and thenrepeatedly measure the size of a given feature. The measurementrepeatability is determined from the standard deviation of the acquireddata. Long term reproducibility, also called precision, is determined ina similar way to static repeatability except that between eachmeasurement the sample is removed from the instrument for an arbitrarylength of time ranging from seconds to days. Unfortunately, therepeatability and reproducibility of a measurement is meaningless if themeasurement is wrong. Accuracy must also be considered. Theabove-described methods do not evaluate the accuracy of an instrumentapart from ensuring proper magnification by calibration with pitchstandards. The reason, in part, that accuracy is not considered is thataccepted accuracy standards are generally not available because thespeed at which semiconductor technology advances usually makes anystandard obsolete very quickly. The result of these methodologies isthat a measurement system under test may be misleadingly denoted astrustworthy.

One proposed solution for metrology instrument assessment introduces newparameters related to accuracy in addition to precision. See Banke andArchie, “Characteristics of Accuracy for CD Metrology,” Proceedings ofSPIE, Volume 3677, pp. 291-308 (1999). This approach deviates from usingstandard product wafers as samples by, for example, constructing wafersreferred to as focus and exposure matrix (FEM) wafers. In thismethodology, the actual CD value is determined for various fields on theFEM by using a respected reference measurement system (RMS). Followingthis approach, the RMS values and measurements from the instrument undertest are compared by a linear regression method that is valid forsituations where both variables are subject to error. Use of the FEMwafers is advantageous because they provide examples of productvariation that under normal manufacturing line circumstances may occuronly after a considerable time has passed. Important parameters of thismethodology include the regression slope, the average offset, and a“poorness-of-fit” parameter called nonlinearity. Despite the existenceof this suite of parameters for repeatability, reproducibility andaccuracy, however, an evaluator must still determine, somewhatarbitrarily, how to combine these various parameters to assess oroptimize an instrument.

In view of the foregoing, there is a need in the art for improvedmethods of assessing and optimizing metrology instruments.

SUMMARY OF THE INVENTION

The invention relates to methods for assessing and optimizing metrologyinstruments by determining a total measurement uncertainty (TMU) basedon precision and accuracy. The TMU is calculated based on a linearregression analysis and removing a reference measuring systemuncertainty (U_(RMS)) from a net residual error. The invention improvesthe TMU analysis by providing improved methods for determining theU_(RMS) under a variety of situations. The TMU provides an objective andmore accurate representation of whether a measurement system under testhas an ability to sense true product variation. The invention alsoincludes a method for determining an uncertainty of the TMU.

The foregoing and other features of the invention will be apparent fromthe following more particular description of embodiments of theinvention.

BRIEF DESCRIPTION OF THE DRAWINGS

The embodiments of this invention will be described in detail, withreference to the following figures, wherein like designations denotelike elements, and wherein:

FIG. 1 shows a graph of data for a measurement system under test versusa reference measurement system.

FIGS. 2A-B show flow diagrams of assessment method embodiments of theinvention.

FIG. 3 shows multiple cross-sectional views of an artifact formeasurement.

FIG. 4 shows a graph of data for a couple of CD scanning electronmicroscopes (SEM) under test versus an atomic force microscope (AFM)reference measurement system.

FIG. 5 shows AFM images for one feature of an artifact.

FIG. 6 shows a graph of variation in feature height and sidewall angleacross the various features through photolithographic stepper focus anddose.

FIGS. 7A-B show flow diagrams of optimization method embodiments of theinvention.

FIG. 8 shows a graph of total measurement uncertainty and correctedprecision versus an amount of SEM data smoothing from the optimizationprocess shown in FIGS. 7A-B.

FIG. 9 shows a flow diagram of a method of determining an uncertainty ina total measurement uncertainty.

DETAILED DESCRIPTION OF THE INVENTION

The description includes the following headings for clarity purposesonly: I. Data Analysis, II. Assessment Method, III. Optimization Method,IV. Improved Reference Measurement System Uncertainty Calculation, V.Uncertainty of Total Measurement Uncertainty (TMU), VI. General TMUUncertainty Estimate for Preferred RMS Uncertainty Estimate, and VII.Conclusion. It should be recognized that while particular types ofmeasurement systems will be mentioned throughout the description thatthe teachings of the invention are applicable to any type of measurementsystem. As will be described below, the discussion of calculation ofU_(RMS) of Section IV can be substituted for the calculation methodsdescribed in sections I-III.

I. Data Analysis

In order to determine a total measurement uncertainty (hereinafter“TMU”) of a measurement system under test (hereinafter “MSUT”), it isnecessary to compare measurement data sets of a MSUT and a referencemeasurement system (hereinafter “RMS”). A conventional technique forcomparing such data sets is linear regression derived by plotting thedata sets against one another as shown in FIG. 1. The following dataanalysis is derived from the paper “Characteristics of Accuracy for CDMetrology,” Proceedings of SPIE, Volume 3677, pp. 291-308 (1999) byBanke and Archie, which describes a form of linear regression upon whichthe invention draws. As used herein, precision shall be referred as aone sigma (σ) value.

When regressing one variable onto another an assumption is made aboutthe relationship between the two variables. Referring to FIG. 1, it isassumed that a MSUT, e.g., a CD SEM, should behave linearly to the firstorder when compared to a set of reference standards, i.e., those from anRMS, e.g., a CD AFM. Such a model would be represented by a slope, β,and an intercept, α, like the following equation:y _(i) =α+βx _(i)+ε_(i)  (1)where y_(i) and x_(i) represent the i^(th) dependent and independentvariables, respectively, and ε_(i) is the i^(th) deviation, or residual,from the model. In terms of the metrology instrument assessment andoptimization data analysis and methods, as discussed in more detailbelow, the independent variable x refers to the MSUT and the dependentvariable y refers to the RMS.

The ordinary least-squares (hereinafter “OLS”) fit is one type ofgeneral linear regression analysis, in which no error is assumed in theindependent variable (MSUT). However, there are situations, especiallyin the case of semiconductor industry metrology applications, where thisassumption is not valid. There are criteria that give some indication asto when, or under what conditions, it is permissible to use the OLS. Onecriterion is based upon the precision of the independent variable,σ_(x), being small compared to the standard deviation of all the xvalues: $\begin{matrix}{\frac{\sigma_{{all}\quad x\quad{values}}}{\sigma_{x}}\operatorname{>>}1} & (2)\end{matrix}$Another criterion for acceptable use of the OLS fit is: $\begin{matrix}{{\beta } \times \frac{\sigma_{x}}{\sigma_{y}}{\operatorname{<<}1}} & (3)\end{matrix}$

If the estimated slope is approximately unity, it is easy to see thatthe precision in the independent measurement (MSUT) must be muchsmaller, or better, than the precision in the dependent variable (RMS)for the OLS to be valid. Perhaps most important in testing the accuracyof an unknown MSUT is the effect of the uncertainty in the referencestandards on the resultant parameters that are used to assess thisaccuracy. To account for this, a method of linear regression thataddresses errors in the y (RMS) and x (MSUT) variables and estimates theslope and intercept of the resultant best-fit line is necessary tofairly evaluate the accuracy of a measurement system.

The Mandel linear regression, as introduced in 1964 and revised in 1984by John Mandel, provides a methodology of handling the least-squares fitwhen both variables are subject to error. One of the benefits of thismore generalized regression analysis is that it can be used in alldegrees of error in x and y, even the case when errors in x are zero,σ_(x)=0. One parameter affecting the Mandel method is a variable λ(referred to herein as the “ratio variable”), which is defined by:$\begin{matrix}{\lambda = \frac{\sigma_{y}^{2}}{\sigma_{x}^{2}}} & (4)\end{matrix}$where σ_(y) and σ_(x) are the precisions of the y (RMS) and x (MSUT)measurements, respectively. In the Mandel method, it is important torecognize that these precisions are based on replication only, notaccuracy. According to the invention, the ratio variable λ is re-definedas: $\begin{matrix}{\lambda = \frac{U_{RMS}^{2}}{U_{MSUT}^{2}}} & (5)\end{matrix}$where U_(RMS) is an RMS “uncertainty” defined as an RMS precision(σ_(RMS)) or an independently determined RMS total measurementuncertainty (TMU_(RMS)), and U_(MSUT) is an MSUT “uncertainty” definedas a corrected precision of the MSUT or a TMU of the MSUT, as will bemore fully described below. The TMU_(RMS) can be determined using themethods as described herein applied to the RMS, i.e., treating the RMSas an MSUT. Unless denoted “TMU_(RMS),” “TMU” shall refer to the TMU forthe MSUT.

The intent of the Mandel method is to start the analysis of the fittingprocedure with some measure of the confidence level for eachmeasurement. A key metric resulting from this regression is the slope ofthe best-fit line: $\begin{matrix}{\hat{\beta} = \frac{S_{yy} - {\lambda\quad S_{xx}} + \sqrt{\left( {S_{yy} - {\lambda\quad S_{xx}}} \right)^{2} + {4\lambda\quad S_{xy}^{2}}}}{2S_{xy}}} & (6)\end{matrix}$where the S_(xx), S_(yy), and S_(xy) are the sum of the squares from theraw data as defined by: $\begin{matrix}{{S_{xx} = {\sum\limits_{i = 1}^{N}\left( {x_{i} - \overset{\_}{x}} \right)^{2}}},{S_{yy} = {\sum\limits_{i = 1}^{N}\left( {y_{i} - \overset{\_}{y}} \right)^{2}}},{S_{xy} = {\sum\limits_{i = 1}^{N}{\left( {x_{i} - \overset{\_}{x}} \right)\left( {y_{i} - \overset{\_}{y}} \right)}}}} & (7)\end{matrix}$where N is the number of ordered pairs. In the general linear regressioncase, where OLS is valid, the uncertainty of the independent variable(MSUT) goes to zero and the ratio variable λ→∞. The estimate for theslope as the ratio variable λ approaches infinity is S_(xy)/S_(xx) andwhen all the error is in the x (MSUT) measurement compared to the y(RMS) measurement, the ratio variable λ approaches zero and the estimatefor the slope is S_(yy)/S_(xy). This would be like regressing x onto y,which points out another feature of the Mandel method of regression. Theanalysis is symmetrical with the x and y variables such that it does notmatter whether x is regressed on y, or y is regressed on x.

Another metric resulting from this methodology is the correctedprecision of a metrology instrument, which is defined as follows:Corrected Precision≡{circumflex over (β)}σ_(x)  (8)As defined, a smaller slope {circumflex over (β)} implies a greaterchange in MSUT measurement for a given change in the RMS values. Use ofa corrected precision is useful because a MSUT could exhibit a smaller(better) precision than other tools under test, yet have a larger(worse) slope. A larger slope would imply a less sensitive measurementtool, while on the other hand, a smaller precision would indicate a moreresolute measurement capable of being sensitive to small changes. Theproduct of these two estimates acts as a balance for the raw,uncorrected, precision. Therefore, for an equivalent corrected precisionof two different MSUTs, a system with a smaller estimated slope{circumflex over (β)} can accommodate a larger precision σ_(x) to yieldan equivalent corrected precision. In other words, the slope correctsthe precision to correspond to the RMS calibrated scale.

As a check and balance on the corrected precision, a specification onthe slope is also required. It is desirable to have a measurement systemwith a unity slope (i.e., slope=1) to maintain a constant offset, whichvaries as a function of the RMS values when the slope is not equal toone. This situation makes for a more complicated correction in amanufacturing environment.

Another parameter of the regression analysis is the estimated intercept,{circumflex over (α)}. This parameter is dependent upon the estimatedslope. As a result, the two parameters of the 1^(st)-order regressionanalysis, i.e., {circumflex over (α)} and {circumflex over (β)}, are notstatistically independent of each other. In addition, since theintercept is a value of y at x=0, it is difficult to get an intuitivemeaning of its value. Instead of this parameter of the regression,another parameter called the offset is used and defined here as:Offset≡Δ={overscore (y)}−{overscore (x)}  (9)where {overscore (x)} and {overscore (y)} are the measurement averagesof a calibration effort. This parameter is independent of the regressionanalysis. Recognizing this and considering that for a calibration efforton a MSUT, its measurements will be regressed against the RMS values,the offset is a reflection of the closeness of the MSUT compared to theRMS.

Another check is that the data needs to be tested to see if the x versusy relationship can be described as linear. This check is completed byconsidering the residual error. The residual error definition isdifferent for the general linear regression (e.g., OLS) case compared tothe Mandel case. The residual error for OLS, d_(i), at each ordered pairof data is defined as:d _(i) =y _(i)−{circumflex over (α)}−{circumflex over (β)}x _(i)  (10)where {circumflex over (α)} and {circumflex over (β)} are the estimatedintercept and slope, respectively, of the OLS regression. The netresidual error D is the square root of the mean-squared error of theseresiduals and can be expressed as: $\begin{matrix}{D^{2} = \frac{\sum\limits_{i = 1}^{N}d_{i}^{2}}{N - 2}} & (11)\end{matrix}$However, this definition of the residual is not correct when the Mandelmethod is applied to the situation of comparing the RMS to the MSUT. Thecorrect net residual error D_(M) is given by: $\begin{matrix}{D_{M} = \sqrt{\frac{\left( {\lambda^{2} + {\hat{\beta}}^{2}} \right)}{\left( {\lambda + {\hat{\beta}}^{2}} \right)^{2}}D^{2}}} & (12)\end{matrix}$The net residual error D_(M) is comprised of both systematic and randomcomponents of error. The method of data gathering and analysis describedherein includes accessing the random component of error by replication,creating essentially a precision estimate. Given precision estimatesσ_(x) and σ_(y) for the x (MSUT) and y (RMS) variables, respectively, itis possible to make an estimate of the input variance of the data set:Var(input)=σ_(y) ²+{circumflex over (β)}²σ_(x) ².  (13)

The slope is included in the above definition for reasons similar to itsintroduction into the corrected precision parameter. The ratio of thesquare of the Mandel net residual error D_(M) to the input variance is aparameter that distinguishes systematic error from random error in thedata set. This quantity is referred to herein as the “nonlinearity”parameter:Nonlinearity=D _(M) ² /Var(input)  (14)When the nonlinearity can be shown to be statistically significantlygreater than unity, then the regression is revealing that the datacontains significant nonlinear systematic behavior.

The invention determines a metric referred to herein as “totalmeasurement uncertainty” (hereinafter “TMU”) that summarizes, in aformat directly comparable to measurement requirements, how well theMSUT measures even if its measurements are corrected by the regressionslope {circumflex over (β)} and intercept {circumflex over (α)}. The TMUmetric can be derived from the general linear regression metrics, orpreferably from the Mandel metrics. In particular, TMU can be derivedfrom the Mandel net residual error D_(M). The Mandel net residual errorD_(M) contains contributions from the RMS uncertainty (U_(RMS)), theMSUT uncertainty (U_(MSUT)), and any nonlinearity in the relationshipbetween measurements from these instruments. Similarly, the TMU can bederived from the net residual error D for a general linear regression,which contains contributions from the RMS uncertainty, i.e., in thiscase the RMS precision (σ_(RMS)), the MSUT corrected precision, and anynonlinearity in the relationship between measurements from theseinstruments.

Conceptually, the TMU is the net residual error (D_(M) or D) without thecontribution from the RMS uncertainty (U_(RMS)). TMU assigns to the MSUTmeasurement all other contributions. As noted above, the “RMSuncertainty” (U_(RMS)) is defined as the RMS precision or anindependently determined RMS total measurement uncertainty (TMU_(RMS)).That is, in one instance, U_(RMS) may simply be considered the precisionof the RMS (σ_(RMS)), i.e., σ_(RMS) is used as an estimate of the TMUfor the RMS. However, where the RMS has a TMU substantially differentthan its precision, TMU_(RMS) can be input to the ratio variable λ (Eq.5) for determining the Mandel net residual error D_(M) and the TMUdefinition. The TMU_(RMS) may be independently derived for the RMS,i.e., treating the RMS as a MSUT compared to another RMS. Based onabove, TMU for a Mandel linear regression can be defined as:TMU={square root}{square root over (D _(M) ² −U _(RMS) ² )}  (15)where D_(M) is the Mandel net residual error. Similarly, TMU for ageneral linear regression, e.g., OLS, can be defined as:TMU={square root}{square root over (D ² −U _(RMS) ² )}  (16)where D is the net residual error.

It should be recognized, relative to the Mandel linear regression, thatwhen the corrected precision of the MSUT is initially used as the MSUTuncertainty (U_(MSUT)) to calculate ratio variable λ, the subsequentlydetermined TMU value for the MSUT from Eq. 15, in some cases, may besubstantially different from the corrected precision for the MSUT (i.e.,U_(MSUT)). In this case, the linear regression may be repeated with thedetermined TMU value substituted for the corrected precision of the MSUTin the definition of the ratio variable λ (Eq. 5). Similarly, when thesubsequently determined TMU for the MSUT is still substantiallydifferent from the MSUT uncertainty used, the linear regression may berepeated with each new estimate of the TMU substituted for the MSUTuncertainty (U_(MSUT)) in the ratio variable λ (Eq. 5) until sufficientconvergence of the MSUT uncertainty (U_(MSUT)) and TMU is achieved todeclare a self-consistent result.

It should also be recognized that, depending upon the skill with whichthis method is executed and the nature of the measurement techniquesused by the two systems, there may be an undesirable contribution fromthe artifact itself. Properly designed applications of this methodshould minimize or eliminate this contribution.

TMU provides a more correct estimate of the MSUT uncertainty than theprecision estimate alone because it addresses the case where there areerrors due both to precision and accuracy. In contrast, the Mandellinear regression method alone addresses situations where both variablesare only subject to the instrument precisions. Accordingly, TMU is amore objective and comprehensive measure of how the MSUT data deviatesfrom the ideal behavior that would generate a straight-line SL plot inFIG. 1, or the inability of the MSUT to measure accurately. It should berecognized, however, that there are differences between TMU and what isgenerally considered as measurement error, i.e., the quadratic sum ofall possible sources of random and systematic error contributions. Inparticular, systematic errors due to magnification calibration errorsand offset errors are not included in the TMU since, in principle, thesecan be reduced to arbitrarily small contributions given sufficientattention to calibration. TMU represents the limit of what can beachieved for the given type of measurement if sufficient attention ispaid to calibration. As a consequence, it represents a measure of theintrinsic measurement worth of the system.

II. Assessment Method

With reference to FIGS. 2-6, a method and program product for assessinga measurement system under test (MSUT) will be described.

Referring to FIG. 2A, a flow diagram of a method for assessing a MSUTaccording to a first embodiment is shown.

In a first step S1, an artifact for use in assessing the MSUT isprepared. With reference to FIG. 3, “artifact” as used herein shallrefer to a plurality of structures 8 provided on a substrate 16. Anartifact is generated to represent variations in a particularsemiconductor process of interest for the particular MSUT. In oneembodiment, an artifact may be process-stressed samples derived fromactual product. FIG. 3 illustrates exemplary structures for a particularprocess including: an under-exposed structure 10, an ideal structure 12(referred to as the “process of record” (POR) structure), and anoverexposed undercut structure 14. Artifact 8 should be constructed toinclude a fair representation of all of the various scenarios that canarise during manufacturing. The types of artifact provided may varydrastically based on, for example, the type of measurement needingassessment, the manufacturing processes that alter the measurement, andmeasurement parameters that alter the measurement such as temperature,probe damage, manufactured product structure or materials, etc.

Returning to FIG. 2A, at step S2, a critical dimension of artifact 8(FIG. 3) is measured using a reference measurement system (RMS) togenerate an RMS data set. The dimension may include, for example, atleast one of line width, depth, height, sidewall angle, top cornerrounding or any other useful dimension. The RMS is any measuring systemthat is trusted within a particular industry or manufacturing process.The measurement step includes characterizing the artifact(s) andproducing documentation detailing structure location and referencevalues. As part of this step, an RMS uncertainty (U_(RMS)) iscalculated. This calculation may include calculation of an RMS precision(U_(RMS)) according to any now known or later developed methodology,e.g., a standard deviation analysis. Alternatively, this calculation mayinclude calculating a TMU_(RMS) according to the methods disclosedherein. That is, the RMS may be treated as an MSUT and compared toanother RMS.

At step S3, the same dimension is measured using the MSUT to generate anMSUT data set. This step includes conducting a long-term reproducibility(precision) study of the MSUT according to any now known or laterdeveloped methodology. As part of this step, a MSUT precision σ_(MSUT)from the MSUT data set is also calculated according to any now known orlater developed methodology, e.g., a standard deviation analysis.

Referring again to FIG. 1, a plot of data measured by an MSUT in theform of a CD SEM versus an RMS in the form of an AFM is shown. Asdiscussed in the data analysis section above, if a MSUT is a perfectmeasuring tool, the data sets should generate a straight line (SL inFIG. 1) when plotted against one another, i.e., y=x. That is, the lineshould have unity slope and an intercept at 0 as generated by identicaldata points. However, a MSUT is never a perfect measuring tool becauseit and the artifact are subject to the myriad of process variations. Inmost instances, a 0 intercept or unity slope are unlikely and, evenworse, may have peaks or curvature in the data. All of this representsinaccuracy in the MSUT.

Steps S4-S5 (FIG. 2A) represent calculations of a total measurementuncertainty (TMU) of the MSUT according to the above-described dataanalysis. In a first part, step S4, a Mandel linear regression, asdiscussed above, of the MSUT and RMS data sets is conducted. The Mandellinear regression produces the parameters of slope, net residual errorof the MSUT (i.e., the MSUT data set compared to the RMS data set),corrected precision of the MSUT and average offset.

Next, at step S5, TMU is determined according to the formula:TMU={square root}{square root over (D _(M) ² −U _(RMS) ² )}  (17)where D_(M) is the Mandel net residual error (Eq. 12) and U_(RMS) is theRMS uncertainty, i.e., the RMS precision (σ_(RMS)) or an independentlydetermined TMU_(RMS). In other words, a TMU for the MSUT is determinedby removing the RMS uncertainty (U_(RMS)) from the net residual errorD_(M).

At step S6, a determination is made as to whether the determined TMU issubstantially different from the MSUT uncertainty (U_(MSUT)). In a firstcycle of steps S4-S5, the MSUT uncertainty is the corrected precision.In subsequent cycles, the MSUT uncertainty is a previously determinedTMU value of the MSUT. If step S6 results in a YES, as discussed above,the Mandel linear regression may be repeated with the previouslydetermined TMU value substituted for the MSUT uncertainty (step S7)(U_(MSUT)) in the ratio variable λ (Eq. 5). The Mandel linear regressionanalysis is preferably repeated until a sufficient convergence of theMSUT uncertainty (U_(MSUT)) and TMU is achieved to declare aself-consistent result. What amounts to “sufficient convergence” or“substantially different” can be user defined, e.g., by a percentage.

If the determination at step S6 is NO, then the determined TMU value isconsidered the final TMU for the MSUT, i.e., sufficient convergence hasoccurred. Based on the final TMU, an objective assessment of the MSUT isachieved.

Referring to FIG. 2B, a flow diagram of a method for assessing a MSUTaccording to a second embodiment is shown. This embodiment issubstantially similar to the embodiment of FIG. 2A, except that thelinear regression can be any general linear regression, e.g., an OLS. Inthis case, the TMU is defined according to the formula:TMU={square root}{square root over (D ² −U _(RMS) ² )}  (18)where D is the net residual error (Eq. 11) and U_(RMS) is the RMSuncertainty, i.e., the RMS precision (σ_(RMS)) or an independentlydetermined TMU_(RMS). The determined TMU in step S5 is the final TMU.

ASSESSMENT EXAMPLE

Referring to FIG. 4, a graph that compares measurements from two CD SEMs(CD SEM A and CD SEM B) to a respected RMS is shown. The artifact usedwas a focus and exposure matrix (FEM) wafer with the maximum dimensionof an isolated line of resist as the feature of interest. This is aparticularly important geometry and material because it is similar to akey semiconductor processing step that determines the speed with whichtransistors can switch. Hence, tighter and more accurate control at thisstep of manufacturing can produce more computer chips that are extremelyfast and profitable. The RMS in this case was an atomic force microscope(AFM), which was trusted to determine the true CD, namely the maximumlinewidth of the resist.

Ideally this data should lie along a straight line with unity slope andzero offset. The nonlinearity (Eq. 14) parameter characterizes thescatter of the data around the best-fit line. This variance of scatteris normalized so that if all of this variance is due to the randommeasurement variance measured by reproducibility, then the nonlinearityequals unity. In this case, CDSEM A has a nonlinearity of 100 whileCDSEM B has a value of 137. Both are disturbingly large numbers. Thefollowing table derived from this data further illustrates the improvedobjectivity of the TMU parameter. Total Corrected Measurement PrecisionUncertainty [nm] [nm] CDSEM A 1.5 20.3 CDSEM B 1.8 26.1

This example illustrates the severe discrepancy between using precisionas the key roadmap parameter and TMU, which contains precision but alsoincludes contributions from accuracy. In the particular example of theresist-isolated line, the problem is associated with severe resist lossduring the printing process, which can have profound changes to the lineshape, and how well the MSUT measures the desired critical dimension.FIG. 5 shows multiple AFM images for one of the features on this FEMwafer. The AFM image shows edge roughness, top corner rounding, and evenundercut. Referring to FIG. 6, a graph shows the variation in featureheight and sidewall angle across the FEM. On the horizontal axis is thephotolithographic stepper focus setting. Across this FEM, the featureheight changes by a factor of three (3). In addition, there issignificant sidewall angle variation.

III. Optimization Method

An application for the above-described assessment methodology and TMUcalculation lies in the optimizing of a measurement system. Conventionalmethods for optimizing a MSUT would seek measurement conditions andalgorithm settings to minimize the precision and offset of themeasurement. Minimization of TMU as described above, however, provides amore objective and comprehensive determination.

Turning to FIG. 7A, a flow diagram of a method of optimizing an MSUTaccording to a first embodiment is shown. In first step, S1, a structure8 (FIG. 3), i.e., artifact, is provided as described above relative tothe assessment method.

Step S2 (FIG. 7A) includes measuring a dimension of the plurality ofstructures according to a measurement parameter using a referencemeasurement system (RMS) to generate an RMS data set. A “measurementparameter” as used herein, refers to any measurement condition oranalysis parameter that affects the outcome of the measurement that canbe controllably altered. A “measurement parameter” may also include acombination of conditions and parameters or a variation of one of these.Measurement parameters may vary, for example, according to the type ofMSUT. For example, for an SEM, a measurement parameter may include atleast one of: a data smoothing amount, an algorithm setting, a beamlanding energy, a current, an edge detection algorithm, a scan rate,etc. For a scatterometer, a measurement parameter may include at leastone of: a spectra averaging timeframe, a spectra wavelength range, anangle of incidence, area of measurement, a density of selectedwavelengths, number of adjustable characteristics in a theoreticalmodel, etc. For an AFM, a measurement parameter may include at least oneof: a number of scans, a timeframe between scans, a scanning speed, adata smoothing amount, area of measurement, a tip shape, etc. A step ofselecting a measurement parameter(s) (not shown) may also be included inthe optimization method. Subsequently, an RMS uncertainty (U_(RMS)) iscalculated. This calculation may include calculation of an RMS precision(σ_(RMS)) according to any now known or later developed methodology,e.g., a standard deviation analysis. Alternatively, this calculation mayinclude calculating a TMU_(RMS) according to the methods disclosedherein. That is, the RMS may be treated as an MSUT and compared toanother RMS.

In next step, S3, measurement of the same dimension of the plurality ofstructures according to the same measurement parameter using the MSUT ismade to generate an MSUT data set. Subsequently, a precision of the MSUTfrom the MSUT data set is calculated.

Step S4 includes, as described above relative to the assessment method,conducting a Mandel linear regression analysis of the MSUT and RMS datasets to determine a corrected precision of the MSUT, and a net residualerror for the MSUT.

Next, at step S5, TMU is determined according to the formula:TMU={square root}{square root over (D _(M) ² −U _(RMS) ² )}  (19)where D_(M) is the Mandel net residual error (Eq. 12) and U_(RMS) is theRMS uncertainty, i.e., the RMS precision (σ_(RMS)) or an independentlydetermined TMU_(RMS). In other words, a TMU for the MSUT is determinedby removing the RMS uncertainty (U_(RMS)) from the net residual errorD_(M).

At step S6, a determination is made as to whether the determined TMU issubstantially different from the uncertainty for the MSUT (U_(MSUT)). Asnoted above, in a first cycle of steps S4-S5, the MSUT uncertainty isthe corrected precision. In subsequent cycles, the MSUT uncertainty is apreviously determined TMU value of the MSUT. If step S6 results in aYES, as discussed above, the Mandel linear regression may be repeatedwith the previously determined TMU value substituted for the MSUTuncertainty (U_(MSUT)) (step S7) in the ratio variable λ (Eq. 5). TheMandel linear regression analysis is preferably repeated until asufficient convergence of the MSUT uncertainty (U_(MSUT)) and TMU isachieved to declare a self-consistent result. What amounts to“sufficient convergence” or “substantially different” can be userdefined, e.g., as a percentage.

If the determination at step S6 is NO, then the determined TMU isconsidered the final TMU for that measurement parameter, and processingproceeds to step S8.

At step S8, a determination is made as to whether another measurementparameter (e.g., CD SEM smoothing filter adjustment) exists. If step S8results in YES, steps S3 to S7 may be repeated for another measurementparameter. The repeating step may recur for any number of measurementparameters. The resulting data includes a number of TMUs withcorresponding measurement parameter(s) and/or artifact structure(s). Ifstep S8 results in NO, processing proceeds to step S9.

Step S9 includes optimizing the MSUT by determining an optimalmeasurement parameter based on a minimal TMU. In particular, a minimalTMU is selected from a plurality of total measurement uncertainties of acorresponding plurality of measurement parameters. The correspondingmeasurement parameter represents the least imprecise and inaccurateenvironment for using the MSUT.

Referring to FIG. 7B, a flow diagram of a method for optimizing a MSUTaccording to a second embodiment is shown. This embodiment issubstantially similar to the embodiment of FIG. 7A, except that thelinear regression can be any general linear regression, e.g., an OLS. Inthis case, the TMU is defined according to the formula:TMU={square root}{square root over (D ² −U _(RMS) ² )}  (20)where D is the net residual error (Eq. 11) and U_(RMS) is the RMSuncertainty, i.e., the RMS precision (σ_(RMS)) or an independentlydetermined TMU_(RMS). In addition, the determined TMU for a measurementparameter in step S5 is considered the final TMU for that particularmeasurement parameter. Step S6 and S7 are identical to steps S8 and S9relative to the description of FIG. 7A.

OPTIMIZATION EXAMPLE

Referring to FIG. 8, an example derived from optimizing measurementconditions on a CDSEM for a resist isolated line geometry is graphicallyillustrated. The CD SEM starting conditions were those of one of the CDSEMs discussed earlier. While several acquisition conditions andalgorithms settings were optimized in this investigation, the graphshown in FIG. 8 illustrates the consequences of changing the amount ofsmoothing done to the raw CD SEM waveform prior to further algorithmanalysis. In particular, the noise reduction from this smoothing has apositive effect upon reducing the corrected precision. However, from thepoint of view of TMU, the trend is opposite. This suggests that the lossof accuracy in tracking the process changes in the artifact is worsewith greater smoothing as evidenced by this trend dominating the TMU.

IV. Improved Reference Measurement System Uncertainty U_(RMS)Calculation

The above-described methodology for determining total measurementuncertainty (TMU) is proposed as an improved method of assessing andoptimizing metrology instruments. TMU analysis provides the correlationslope, the TMU, and the average offset. While the previous work with TMUanalysis uses a general description of uncertainty of the RMS (U_(RMS)),the known approaches to uncertainty of the RMS are not adequate wherethe measurement uncertainty of the MSUT is reduced to a levelapproaching or below that of the RMS. Thus, improved methods ofcharacterizing the uncertainty of the RMS (U_(RMS)) to enhance theutility of TMU analyses are also needed.

The U_(RMS) is made up of both systematic and random error components.The systematic errors are often difficult to determine. Those systematicerrors that can be determined should be removed from the U_(RMS) (e.g.,by tool calibration). A proper construction of the RMS assumes that thesystematic errors have been removed. Accordingly, improvements indetermining U_(RMS) should focus on just the random error component ofthe U_(RMS). If the letter “V” represents a variance, the randomcomponents of the U_(RMS) include:

-   -   1. The short term precision/repeatability component V_(ST);    -   2. The across grating variation component V_(AG);    -   3. The “multiple reference measurement system” component        V_(MRMS); and    -   4. The long term precision/repeatability component V_(LT).        Although V_(AG) is not technically a part of the U_(RMS), TMU        analysis dictates that it must be associated either with the        MSUT or the RMS. Because the TMU is meant to be a measure of how        well the MSUT measures, V_(AG) is assigned to the U_(RMS).

The U_(RMS), when expressed as a variance, may be just

-   -   V_(URMS)=V_(ST)+V_(AG), or may include all of the variances as:    -   V_(URMS)=V_(ST)+V_(AG)+V_(MRMS)+V_(LT).

A. Short Term Precision and Across-Grating Variation Components

It is often best to determine the short term precision andacross-grating variation components together without separating thembecause, when making a reference measurement using non-unique patternrecognition, both of these components affect the measurement. In orderto determine them, three questions must be answered:

-   -   a) How will the data be compared (by chip average, by wafer        average, or by lot average)?    -   b) How many measurements per grating were made (one or        multiple)?    -   c) How does the process vary from one wafer to the next?

There are three common ways that data can be compared in a correlationplot. The first way involves averaging all RMS measurements across agrating and plotting these averages against the grating averages fromthe MSUT. Usually only one grating of a given type is measured per chip;in this case, this type of correlation plot is called a “by chip” plot.If multiple identical gratings are measured on each chip, each gratingcould be considered a separate chip; this correlation plot would stillbe considered a “by chip” plot. When all measurements from a gratingtype are averaged across a wafer so that each point in the plotrepresents a wafer average, the correlation plot is called a “by wafer”plot. Similarly, averaging measurements across a lot yields a “by lot”comparison. Different comparisons are useful in different ways. Forexample, “by wafer” comparisons are more useful than the others whendoing wafer feedforward analysis. “By lot” comparisons may be best inmanufacturing if the wafers in a lot measured by the RMS are notnecessarily the same as those measured by the MSUT.

The second consideration when determining V_(ST)+V_(AG) is how manymeasurements per grating were made. The method for determiningV_(ST)+V_(AG) is fundamentally different depending on whether one ormultiple measurements are made. However, this method does not depend onwhether the RMS uses a microscopic sampling technique (e.g., a CD-SEM orAFM) or a macroscopic sampling technique (e.g., a scatterometer). Whenmultiple measurements are made, the microscopic sampling should bespread across the grating in order to properly capture theacross-grating variation component. Multiple measurements (two or more)are preferred, as the associated calculation is a little easier, and theresulting estimate for V_(ST)+V_(AG) is more accurate.

The last consideration depends on the wafer processing. As will bedescribed below, wafers that were nominally processed the same for allprocesses that may affect the measurements can be grouped together toaid in the calculation of V_(ST)+V_(AG). Of course, no two wafers areprocessed exactly the same due to subtle, uncontrollable processvariations, but the more similarly they are processed, the better thedetermination of V_(ST)+V_(AG). Process variations can be caused, forexample, by intentional variations in the etch process or film stack.When evaluating a process variation to determine whether it issignificant, the critical question to ask is: “Does this process affectthe measurements across the wafer as it is varied from one wafer to thenext?” For example, etch process “A” may have better center-to-edgeuniformity than etch process “B.” This means that wafers processed withone of these etches should be conceptually separated from wafersprocessed with the other etch. The reasoning for this will be explainedbelow.

Answers to the three questions above are used to determine howV_(ST)+V_(AG) is calculated. But before this calculation is described,the variables that are used in the calculation need to be defined, whichare shown in Table 1. Table 2 summarizes methods for determining thecomponent V_(ST)+V_(AG) of the U_(RMS) for many situations.

Consider case #1. Here, a “by chip” analysis is desired, and multiplemeasurements have been collected from the grating using the RMS. Theshort term precision and across-grating components can be calculated byfirst determining the variance of all the measurements made on grating“i.” A variance is calculated for each grating, ending with a total ofN_(C) variances (one for each chip). This assumes that only one gratingof a given type per chip is measured. These variances are then averaged(hence the summation and the division by N_(C)). The average variance isthen divided by n_(G), the number of measurements per grating. This isdone because the more measurements that are collected across thegrating, the smaller the uncertainty in the RMS becomes. Note that case#1 is independent of the variation of wafer processing because it isassumed that the wafer processing does not affect the across-gratingvariation. TABLE 1 Definition of variables used to calculate V_(ST) +V_(AG). S_(Gi) = set of all measurements across grating i {overscore(y)}_(Wj) = average of all measurements across wafer j S_(DCk) = set ofall measurement differences from the wafer average {overscore (y)}_(Wj)for chip k from all wafers V(s) = variance of the set s n_(G) = numberof measurements/grating n_(C) = number of measured chips/wafer n_(W) =number of measured wafers/lot N_(L) = total number of lots N_(C) = n_(C)× n_(W) × N_(L) = total number of measured chips N_(W) = n_(W) × N_(L) =total number of measured wafers

Case #2 also involves “by chip” analysis, but now only one measurementper grating was made. Determining the variance of a single measurementis impossible, so the across-grating variation must be captured in adifferent way. To do this, measurements from different gratings must becarefully combined into sets so that across-wafer and wafer-to-wafervariation is not inadvertently captured. To avoid capturing across-wafervariation, only the measurements from the same chip across all wafersare grouped together. Taking the variance of each of these sets ofmeasurements, however, still captures wafer-to-wafer variation. To avoidthis, the average of all measurements across a wafer is calculated. Forwafer “j” this average would be {overscore (y)}_(Wj). Each measurementis then subtracted from its wafer average. The set of all of thesemeasurement differences for chip “k” from all wafers is called s_(DCk).Thus, taking the variance of s_(DCk) avoids both across-wafer andwafer-to-wafer variation. The number of these variances equals thenumber of measured chips per wafer, n_(C). Averaging these n_(C)variances then determines V_(ST)+V_(AG). This methodology essentially“pretends” that the measurements in each set s_(DCk) come from the exactsame grating. If the number of measured wafers, N_(W), is one, each setincludes only one number. Since it is meaningless to take the varianceof these single-number sets, V_(ST)+V_(AG) must instead be calculated ina separate exercise. It is noted that for case #2 there is no variationin the wafer processing. If there was such a variation, then chip “k” onone wafer may be treated differently relative to the wafer average thanchip “k” on another wafer from a different process. This would addvariation to s_(DCk) that would depend on the wafer process, resultingin an artificially high variance.

Case #3 is similar to case #2, except that now there is some variationin the wafer processing. What is meant by this is that the wafers arenow grouped so that wafers within a group were processed the same, buteach group was processed differently. The groups need to have more thanone wafer each in order for this to be considered case #3. Themethodology used for case #2 is now used, but each wafer group is keptseparate. That is, each set s_(DCk) only spans measurements acrosswafers within a single group. One estimate of V_(ST)+V_(AG) for eachwafer group is then calculated. Using each of these estimates, aweighted average, according to the number of wafers in each group, isthen computed to determine the final value of V_(ST)+V_(AG).

Case #4 represents the situation where each wafer group includes onlyone wafer. Here, it is said that there is complete variation in thewafer processing. That is, each wafer is processed uniquely. When thisoccurs, each s_(DCk) includes only one measurement, so the methodologydescribed in cases #2 and #3 breaks down. Therefore, V_(ST)+V_(AG) mustbe determined in a separate exercise.

Cases 5-8 are analogous to cases 1-4, except now the desired analysis isby wafer. The only difference in the calculation now is the extradivision by n_(C), the number of measured chips per wafer. Thiscalculation is completed because the more chips that are measured perwafer, the better the determination of the U_(RMS).

Cases 9-12 are used for “by lot” analysis, and are analogous to cases5-8. The factor (1/n_(W)) has been included because the more wafers thatare measured per lot, the better the determination of the U_(RMS). TABLE2 Strategies for calculating V_(ST) + V_(AG), depending on intendedanalysis, sampling plan, and wafer processing. Variation of waferprocess conditions: By chip, One or 1. none (constant) By wafer,multiple 2. some (wafer groups) Case or meas./ 3. complete (each # Bylot grating wafer unique) V_(ST) + V_(AG) 1 By chip multiple none, some,or complete$\frac{1}{n_{G}N_{C}}{\sum\limits_{i = 1}^{N_{C}}{V\left( s_{Gi} \right)}}$2 By chip one none $\begin{matrix}{\frac{1}{n_{C}}{\sum\limits_{k = 1}^{n_{C}}{V\left( s_{NCk} \right)}}} \\{{{{If}\quad N_{W}} = 1},{{{determine}\quad V_{ST}} + {V_{AG}\quad{{separately}.}}}}\end{matrix}\quad$ 3 By chip one some Use results from case #2, exceptkeep each wafer group separate. After determining the variance for eachwafer group, the weighted average (according to # of wafers in eachgroup) of these variances determines V_(ST) + V_(AG). 4 By chip onecomplete Determine V_(ST) + V_(AG) separately. 5 By wafer multiple none,some, or complete$\frac{1}{n_{C}n_{G}N_{C}}{\sum\limits_{i = 1}^{N_{C}}{V\left( s_{Gi} \right)}}$6 By wafer one none$\frac{1}{n_{C}^{2}}{\sum\limits_{k = 1}^{n_{C}}{V\left( s_{NCk} \right)}}$7 By wafer one some Use results from case #6, except keep each wafergroup separate. After determining the variance for each wafer group, theweighted average (according to # of wafers in each group) of thesevariances determines V_(ST) + V_(AG). 8 By wafer one complete DetermineV_(ST) + V_(AG) separately. 9 By lot multiple none, some, or complete$\frac{1}{n_{W}n_{C}n_{G}N_{C}}{\sum\limits_{i = 1}^{N_{C}}{V\left( s_{Gi} \right)}}$10 By lot one none$\frac{1}{n_{W}n_{C}^{2}}{\sum\limits_{k = 1}^{n_{C}}{V\left( s_{NCk} \right)}}$11 By lot one some Use results from case #10, except keep each wafergroup separate. After determining the variance for each wafer group, theweighted average (according to # of wafers in each group) of thesevariances determines V_(ST) + V_(AG). 12 By lot one complete DetermineV_(ST) + V_(AG) separately.

Special instances exist where the wafer processing may appear to becategorized as “some” or “complete,” but is in fact “none.” Theseinstances are based on chip-level processing (e.g., lithography) ratherthan wafer-level processing (e.g., etch). One such instance is wheredifferent wafers receive different lithography doses or foci, but thedose and focus for each wafer is constant across that wafer. Exposingwafers at these different conditions is not expected to affectmeasurements across the wafer because these are chip-level processingconditions. Thus, any across-wafer variation differences between wafersexposed with a different dose or focus is not expected to be caused bythese exposure conditions. Across-wafer variation differences wouldinstead be caused by intentional wafer-level processing differences(e.g., etch) or by subtle, uncontrollable wafer-to-wafer processingdifferences.

Note that the methods described in table 2 use the referencemeasurements themselves to determine V_(ST)+V_(AG). The advantage tothis is that no additional measurements need to be made. However, whenV_(ST)+V_(AG) must be determined separately, additional measurementsbeyond the reference measurements are required. For this less-preferredmethod, multiple measurements across one or more gratings can be made.Then, the methods shown in case #1, #5, or #9 must be used.

B. “Multiple Reference Measurement Systems” (MRMS) Component

This variance component must be considered when more than one referencemeasurement system has been used in a correlation plot. By this, it ismeant that the reference measurement systems are of the same type ormodel (e.g., the same type of CD-SEM or AFM). A composite RMS (e.g.,CD-SEM, which has been calibrated by CD-AFM, which has been calibratedby TEM) is different and is not what is meant by “multiple referencemeasurement systems.”

The best way to determine the MRMS component is to conduct a TMUcalibration exercise between the RMS tools using either one or morewafers from the wafer set that is used for the TMU analysis, or usingone or more wafers that are representative of the wafers used for theTMU analysis (i.e., the same application). Wafers used in this way arecalled “calibration wafers.” Then multiple measurements across thewafer(s) of the exact same feature (using unique pattern recognition)must be made using all reference measurement systems used in thecorrelation plot. One system must then be identified as the “goldensystem.” This can be any of the reference measurement systems, and doesnot have to be what is considered the “best” system. The easiest choicefor the golden system is the one that is used most frequently in thecorrelation plot. Finally, using TMU analysis, all measurements fromnon-golden systems should be converted to golden-system-equivalentmeasurements. This does not determine V_(MRMS) per se, but does properlytake into consideration the use of multiple reference measurementsystems, and eliminates the need to calculate an actual value forV_(MRMS).

If it is inconvenient to conduct this calibration exercise for all RMSs,an alternative is offered which provides an estimate for V_(MRMS). Todetermine this estimate, first measure the calibration wafer(s) on twodifferent reference measurement systems; ideally, one should be thegolden system. Use TMU analysis to determine the TMU and average offsetbetween the systems. Note that the analysis should be iterated so thatthe U_(RMS) and TMU are equal at the end of the iterative process. Thisiterative analysis is done because the reference measurement systems arethe same type, and so the uncertainty should be shared equally betweenthem. The next step is to convert the TMU to a variance by dividing bythree, then squaring. This variance will be denoted V_(TMU) _(—)_(MRMS). Next, the average offset must be squared. This number, whichhas the same units as a variance, will be denoted V_(AVG) _(—) _(MRMS).If it is at least known that all of the other RMSs measure in a“similar” manner to the non-golden system that was calibrated, then itcan be assumed (estimated) that they would have a similar TMU andaverage offset as the calibrated system, when compared to the goldensystem. The accuracy of this estimate relies on how well the other RMSsare matched to the calibrated non-golden system. When this assumption istrue, VMRMS can be estimated asV _(MRMS) =V _(TMU) _(—) _(MRMS) +V _(AVG) _(—) _(MRMS).  (2)

Note that only one V_(MRMS) should be calculated; there is not aV_(MRMS) for each RMS. Because measurements made on the non-goldencalibrated system can be “converted” to golden-system measurements,while equation (2) is helpful when some measurements are made on theuncalibrated systems, care must be used when using this methodology.Different ratios for the number of measurements collected on each typeof RMS affect how this methodology is used. For example, if 10% of themeasurements are collected on the golden system, 80% on the non-goldencalibrated system, and 10% on uncalibrated systems, the methodologycould be implemented differently than if the second and thirdpercentages are switched.

C. Long Term Precision Component

The long term precision component is only relevant and present if thereference measurements have been taken over a long period of time.Furthermore, if multiple RMSs are used, the long term precisioncomponent is only relevant and present if the reference measurementsfrom a given RMS have been taken over a long period of time. That is, ifthe measurements from each RMS are made over a short period of time, butthe time period between when the first and last RMSs are used is long,then the long term precision component is not needed.

To determine V_(LT), two exercises must be completed. The first is todetermine the short term precision component (V_(ST)) directly andindependently of V_(AG). To do this, repeated measurements using uniquepattern recognition must be collected on a relevant structure from thecalibration wafer(s) over a short period of time. Then V_(ST) can becalculated from these data using standard techniques.

The second exercise is identical to the first, except that themeasurements are made over a long period of time (similar in scope tothe period used to collect the actual reference data). Since thesemeasurements are affected by both short and long term precision, thevariance V_(ST) _(—) _(LT) that is derived using standard techniques isjustV _(ST) _(—) _(LT) =V _(ST) +V _(LT).  (3)Since V_(ST) was already determined in the first exercise, V_(LT) caneasily be calculated:V _(LT) =V _(ST) _(—) _(LT) −V _(ST).  (4)

D. Display of TMU Results

The U_(RMS) determination techniques described above are advantageous toachieve an accurate TMU estimate when the TMU is close to or less thanthe U_(RMS). Therefore, given extremely small TMUs, the U_(RMS)methodology may be used in all TMU analyses.

Another improvement to TMU analysis is described by Sendelbach et. al.and involves the use of TMU error bars. M. Sendelbach et al,“Correlating Scatterometry to CD-SEM and electrical gate measurements atthe 90 nm mode using TMU analysis”, Proceedings of SPIE, Vol. 5375,Metrology, Inspection, and Process Control for Microlithography XVIII,Editor, Richard M. Silver, pp. 550-563, 2004. This technique estimatesthe uncertainty with which a TMU result is calculated by providing anupper and lower estimate for TMU. The results shown use this techniquewith TMU uncertainty bounds determined using a confidence level of 90%.

V. Uncertainty of Total Measurement Uncertainty (TMU)

The TMU is derived from two quantities, the measurement uncertainty ofthe reference measurement system U_(RMS) and the net residual errorcoming from the Mandel analysis D_(M), according to the followingequation:σ_(TMU) ={square root}{square root over (D _(M) ² −U _(RMS) ² )}.Both U_(RMS) and D_(M) are measures of the distribution widths of randomvariables which follow normal distributions. Accordingly, theseestimates follow chi square probability distributions.

There are three situations that can arise:

-   -   1. When U_(RMS) is very well determined, then the calculation        proceeds by determining the confidence limits for D_(M) by using        chi-squared distribution tables. The confidence limits for TMU        (σ_(TMU)) are then determined by using the above equation to        relate TMU confidence limits to Mandel confidence limits.    -   2. When large numbers of measurements (e.g., >40 typically) are        used to determine U_(RMS) and D_(M), then the chi-squared        probability distributions are fairly symmetrical and a        propagation of error method can be used. In this situation, the        RMS uncertainty can be stated as U_(RMS)±δ_(RMS) and the Mandel        net residual error uncertainty can be stated as        D_(M)±δ_(Mandel). Then, the TMU uncertainty can be written as        σ_(TMU)±δ_(TMU) where        $\delta_{TMU} = \sqrt{{\frac{D_{M}^{2}}{\sigma_{TMU}^{2}}\delta_{Mandel}^{2}} + {\frac{U_{RMS}^{2}}{\sigma_{TMU}^{2}}\delta_{RMS}^{2}}}$        The δ_(Mandel), δ_(TMU) and δ_(RMS) represent the possible        deviation range for each value.    -   3. The most difficult case is when the number of measurements in        either the determination of U_(RMS) or D_(M) or both is not        large. The chi-squared probability distributions are then        asymmetric and combining the uncertainties cannot be done by        propagation of error.

The situation of case 3 and to a lesser extent, case 2, can also haveanother problem. When the confidence interval for U_(RMS) is large thenthe TMU regression analysis depends on which value in the interval isused. This implies that D_(M) can depend on the choice of U_(RMS). Inother words, a new source of error enters into the determination of TMU.If this new source of error is significant in a particular applicationwhere N is large, then the method described in case 2 cannot be used andthe general solution for case 3 would be needed.

VI. General TMU Uncertainty Estimate for Preferred RMS UncertaintyEstimate

The finite size of the data set used in TMU analysis imposes aninteresting property on the uncertainty of the reference measurementsystem (U_(RMS)) when U_(RMS)=σ_(RMS), the precision of the RMS. As withany subset of the full population of a random variable in a statisticalsense, the standard deviation of that subset is an estimate of thestandard deviation of the full population. The particular episode ofreference data gathering will manifest a particular standard deviationthat is embedded in the Mandel regression output variance {circumflexover (σ)}_(Mandel) ² such that {circumflex over (σ)}_(Mandel) ² is afunction of σ_(RMS) ². Expressed mathematically,{circumflex over (σ)}_(Mandel) ²={circumflex over (σ)}_(Mandel)²(σ_(RMS) ²)The definition of TMUTMU={circumflex over (σ)} _(TMU)={square root}{square root over({circumflex over (σ)})}_(Mandel) ²−σ_(RMS) ²is predicated on σ_(RMS) being an unbiased estimate of the embeddedstandard deviation of the reference measurement system for theparticular episode of reference data gathering. The correlation of thesetwo terms in the above equation indicates that the minimum uncertainty,or the best confidence, in the TMU is obtained when the estimates forthe σ_(RMS) ² and the {circumflex over (σ)}_(Mandel) ² are gatheredsimultaneously. One methodology described below, within this section,achieves this simultaneous sampling objective. The same argument appliesto the precision of the measurement system under test σ_(MSUT) which isneeded to determine another useful quantity often called the relativeaccuracy:Relative_Accuracy={square root}{square root over ({circumflex over(σ)})}_(Mandel) ²−σ_(RMS) ²−σ_(MSUT) ².

One method of gathering an estimate of the RMS uncertainty and theprecision of the MSUT during the full calibration process is to performthe calibration in replicates. That is, perform the gathering of themeasurement data from the RMS and the MSUT in several replicationcycles. Once all of the calibration artifacts (i.e., measurementstructures) have been measured by each measurement system, additionalcycles of measurements are performed randomly arranging the measurementsequence each cycle. There are standard statistical methods to estimatethe RMS and MSUT measurement standard deviation from this data. In thisway, the standard deviations are calculated during the same time thatthe calibration process is executed, which also results in the outputvariance of the regression, {circumflex over (σ)}_(Mandel) ².

Referring to FIG. 9, a flow diagram is shown of a method according tothe present invention of estimating the uncertainty of the estimatedTMU, wherein a single calibration episode is treated as one of manyattempts. In step S1, a plurality of measurements are made, e.g., ofstructures that represent variations in a semiconductor process. In stepS2, a net residual error (D), e.g., a Mandel net residual error (D_(M)),and a reference measurement system uncertainty (σ_(RMS)) is calculatedfor each measurement. In step S3, a variance estimate is calculated foreach D_(M) and σ_(RMS). In step S4, if histograms for these varianceestimates are constructed, they will obey Chi-square (χ²) probabilitydistribution functions (pdf), which are known functions dependent onlyon the number of data used in the calibration. It is well-known thatforming the difference between two random variables obeying Chi-square(χ²) probability distributions produces a random variable that obeys aprobability distribution function that is constructed by convolution ofthe starting two pdfs constrained by the algebraic expression connectingthe three random variables. If the convolution operation is representedby the symbol, {circle over (×)}, then the expression for the pdf of theTMU can be represented aspdf _(TMU)=χ_(D) _(M) ₂ ²{circle over (×)}χ_(σ) _(RMS) ₂ ²where χ_(D) _(M) ₂ ² and χ_(σ) _(RMS) ₂ ² are the pdfs for {circumflexover (σ)}_(Mandel) ² and σ_(RMS) ², respectively. In step S5, such apdf_(TMU) is calculated. It is well known how to examine the pdf of arandom variable to determine a range of the random variable thatcontains a percentage of all attempts. In step S6, the TMU range isdetermined. The percentage is called the confidence level and is chosenarbitrarily by the user. The range determines the uncertainty estimatefor the TMU.

VII. CONCLUSION

In the previous discussion, it will be understood that the method stepsdiscussed may be performed by a processor executing instructions ofprogram product stored in a memory. It is understood that the variousdevices, modules, mechanisms and systems described herein may berealized in hardware, software, or a combination of hardware andsoftware, and may be compartmentalized other than as shown. They may beimplemented by any type of computer system or other apparatus adaptedfor carrying out the methods described herein. A typical combination ofhardware and software could be a general-purpose computer system with acomputer program that, when loaded and executed, controls the computersystem such that it carries out the methods described herein.Alternatively, a specific use computer, containing specialized hardwarefor carrying out one or more of the functional tasks of the inventioncould be utilized. The present invention can also be embedded in acomputer program product, which comprises all the features enabling theimplementation of the methods and functions described herein, andwhich—when loaded in a computer system—is able to carry out thesemethods and functions. Computer program, software program, program,program product, or software, in the present context mean anyexpression, in any language, code or notation, of a set of instructionsintended to cause a system having an information processing capabilityto perform a particular function either directly or after the following:(a) conversion to another language, code or notation; and/or (b)reproduction in a different material form.

While this invention has been described in conjunction with the specificembodiments outlined above, it is evident that many alternatives,modifications and variations will be apparent to those skilled in theart. Accordingly, the embodiments of the invention as set forth aboveare intended to be illustrative, not limiting. Various changes may bemade without departing from the spirit and scope of the invention asdefined in the following claims.

1. A method for assessing a measurement system under test (MSUT), themethod comprising the steps of: (a) providing a substrate having aplurality of structures; (b) measuring a dimension of the plurality ofstructures using a reference measurement system (RMS) to generate afirst data set, and calculating an RMS uncertainty (U_(RMS)) from thefirst data set, where the RMS uncertainty (U_(RMS)) is defined as one of(i) an RMS precision; (ii) an independently determined RMS totalmeasurement uncertainty (TMU_(RMS)); and (iii) V_(U) _(RMS)=V_(ST)+V_(AG), wherein V_(U) _(RMS) is U_(RMS) expressed as a variance,V_(ST) is a short term precision variance, and V_(AG) is an acrossgrating variance; (c) measuring the dimension of the plurality ofstructures using the MSUT to generate a second data set, and calculatinga precision of the MSUT from the second data set; (d) conducting alinear regression analysis of the first and second data sets todetermine a corrected precision of the MSUT and a net residual error;and (e) determining a total measurement uncertainty (TMU) for the MSUTby removing the RMS uncertainty (U_(RMS)) from the net residual error.2. The method of claim 1, wherein the plurality of structures representvariations in a semiconductor process.
 3. The method of claim 1, whereinthe dimension includes at least one of line width, depth, height,sidewall angle and top corner rounding.
 4. The method of claim 1,wherein the TMU for the MSUT is determined according to the formula:TMU={square root}{square root over (D ² −U _(RMS) ² )} where D is thenet residual error.
 5. The method of claim 1, wherein the linearregression is calculated using a Mandel linear regression wherein aratio variable λ is defined according to the formula:$\lambda = \frac{U_{RMS}^{2}}{U_{MSUT}^{2}}$ where U_(MSUT) is as anMSUT uncertainty defined as one of the corrected precision of the MSUTand the TMU for the MSUT.
 6. The method of claim 5, wherein, in the casethat the TMU for the MSUT is substantially different than the MSUTuncertainty (U_(MSUT)) after step (e), steps (d) and (e) are repeatedusing the TMU for the MSUT as the MSUT uncertainty (U_(MSUT)) indetermining the ratio variable λ.
 7. The method of claim 5, wherein theTMU for the MSUT is determined according to the formula:TMU={square root}{square root over (D _(M) ² −U _(RMS) ² )} where D_(M)is the Mandel net residual error.
 8. The method of claim 1, whereinV_(U) _(RMS) further includes at least one of a multiple referencesystem variance (V_(MRMS)) and a long term precision variance V_(LT)according to the equationV _(U) _(RMS) =V _(ST) +V _(AG) +V _(MRMS) +V _(LT).
 9. The method ofclaim 1, further comprising the step of determining an uncertainty inthe total measurement uncertainty (TMU), the method comprising the stepsof: determining a confidence limit for a Mandel net residual error(D_(M)); and determining a confidence limit for the total measurementuncertainty (σ_(TMU)) according to the equationσ_(TMU) ={square root}{square root over (D _(M) ² −U _(RMS) ² )}, whereU_(RMS) is a measurement uncertainty of the reference measurementsystem.
 10. The method of claim 1, further comprising the step ofestimating an uncertainty in an estimated total measurement uncertainty(TMU), comprising the steps of: making a plurality of measurements;calculating a Mandel net residual error (D_(M)) and a referencemeasurement system uncertainty (σ_(RMS)) for each measurement;calculating a variance for each D_(M) and σ_(RMS); constructing aChi-squared probability distribution function (χ²-pdf) for the D_(M)variances and the σ_(RMS) variances; calculating a χ²-pdf for the TMUaccording to the equationpdf _(TMU)=χ_(D) _(M) ₂ ²{circle over (×)}χ_(σ) _(RMS) ₂ ², whereinpdf_(TMU) is the TMU χ²-pdf, χ_(D) _(M) ₂ ² is the D_(M) variance χ²-pdfχ_(σ) _(RMS) ₂ ² is the arms variance χ²-pdf, and {circle over (×)} is aconvolution operation; and determining a TMU range from pdf_(TMU),wherein the TMU range determines the uncertainty in the estimated TMU.11. The method of claim 1, wherein each of D_(M) and U_(RMS) isdetermined by at least about 40 measurements and wherein σ_(TMU) isrepresented as σ_(TMU)±δ_(TMU); U_(RMS) is represented asU_(RMS)±δ_(RMS); D_(M) is represented as D_(M)±δ_(Mandel); and theequation σ_(TMU)={square root}{square root over (D_(M) ²−U_(RMS) ² )} isrepresented as$\delta_{TMU} = {\sqrt{{\frac{D_{M}^{2}}{\sigma_{TMU}^{2}}\delta_{Mandel}^{2}} + {\frac{U_{R\quad{MS}}^{2}}{\sigma_{TMU}^{2}}\delta_{R\quad{MS}}^{2}}}.}$12. A method for optimizing a measurement system under test (MSUT), themethod comprising the steps of: (a) providing a plurality of structures;(b) measuring a dimension of the plurality of structures according to ameasurement parameter using a reference measurement system (RMS) togenerate a first data set, and calculating an RMS uncertainty (U_(RMS))from the first data set, where the RMS uncertainty (U_(RMS)) is definedas one of (i) an RMS precision; (ii) an independently determined RMStotal measurement uncertainty (TMU_(RMS)); and (iii) V_(U) _(RMS)=V_(ST)+V_(AG), wherein V_(U) _(RMS) is U_(RMS) expressed as a variance,V_(ST) is a short term precision variance, and V_(AG) is an acrossgrating variance; (c) measuring the dimension of the plurality ofstructures according to the measurement parameter using the MSUT togenerate a second data set, and calculating a precision of the MSUT fromthe second data set; (d) conducting a linear regression analysis of thefirst and second data sets to determine a corrected precision of theMSUT and a net residual error; (e) determining a total measurementuncertainty (TMU) for the MSUT by removing the RMS uncertainty (U_(RMS))from the net residual error; (f) repeating steps (c) to (e) for at leastone other measurement parameter; and (g) optimizing the MSUT bydetermining an optimal measurement parameter based on a minimal totalmeasurement uncertainty.
 13. The method of claim 12, further comprisingthe step of selecting a set of measurement parameters to be evaluated.14. The method of claim 12, wherein the MSUT is an SEM and a measurementparameter includes at least one of: a data smoothing amount, analgorithm setting, a beam landing energy, a current, an edge detectionalgorithm and a scan rate.
 15. The method of claim 12, wherein the MSUTis a scatterometer and a measurement parameter includes at least one of:a spectra averaging timeframe, a spectra wavelength range, an angle ofincidence and area of measurement, a density of selected wavelengths anda number of adjustable characteristics in a theoretical model.
 16. Themethod of claim 12, wherein the MSUT is an AFM and a measurementparameter includes at least one of: a number of scans, a timeframebetween scans, a scanning speed, a data smoothing amount and area ofmeasurement, and a tip shape.
 17. The method of claim 12, wherein theplurality of structures represent variations in a semiconductor process.18. The method of claim 12, wherein the dimension includes at least oneof line width, depth, height, sidewall angle and top corner rounding.19. The method of claim 12, wherein V_(U) _(RMS) further includes atleast one of a multiple reference system variance (V_(MRMS)) and a longterm precision variance V_(LT) according to the equationV _(U) _(RMS) =V _(ST) +V _(AG) +V _(MRMS) +V _(LT).
 20. The method ofclaim 12, wherein a total measurement uncertainty (TMU) for the MSUT isdetermined according to the formula:TMU={square root}{square root over (D ² −U _(RMS) ² )} where D is thenet residual error.
 21. The method of claim 12, wherein the linearregression is calculated using a Mandel linear regression wherein aratio variable λ is defined according to the formula:$\lambda = \frac{U_{R\quad{MS}}^{2}\quad}{U_{MSUT}^{2}}$ where U_(MSUT)is as an MSUT uncertainty defined as one of the corrected precision ofthe MSUT and the TMU for the MSUT.
 22. The method of claim 21, wherein,in the case that the TMU for the MSUT is substantially different thanthe MSUT uncertainty (U_(MSUT)) after step (e), steps (d) and (e) arerepeated using the TMU for the MSUT as the MSUT uncertainty (U_(MSUT))in determining the ratio variable λ.
 23. The method of claim 12, furthercomprising the step of determining an uncertainty in the totalmeasurement uncertainty (TMU), the method comprising the steps of:determining a confidence limit for a Mandel net residual error (D_(M));and determining a confidence limit for the total measurement uncertainty(σ_(TMU)) according to the equationσ_(TMU) ={square root}{square root over (D _(M) ² −U _(RMS) ² )}, whereU_(RMS) is a measurement uncertainty of the reference measurementsystem.
 24. The method of claim 12, further comprising the step ofestimating an uncertainty in an estimated total measurement uncertainty(TMU), comprising the steps of: making a plurality of measurements;calculating a Mandel net residual error (D_(M)) and a referencemeasurement system uncertainty (σ_(RMS)) for each measurement;calculating a variance for each D_(M) and σ_(RMS); constructing aChi-squared probability distribution function (χ²-pdf) for the D_(M)variances and the σ_(RMS) variances; calculating a χ²-pdf for the TMUaccording to the equationpdf _(TMU)=χ_(D) _(M) ₂ ²{circle over (×)}χ_(σ) _(RMS) ₂ ², whereinpdf_(TMU) is the TMU χ²-pdf, χ_(D) _(M) ₂ ² is the D_(M) varianceχ²-pdf, χ_(σ) _(RMS) ₂ ² the σ_(RMS) variance χ²-pdf, and {circle over(×)} is a convolution operation; and determining a TMU range frompdf_(TMU), wherein the TMU range determines the uncertainty in theestimated TMU.
 25. A method for estimating an uncertainty in anestimated total measurement uncertainty (TMU) comprising the steps of:making a plurality of measurements; calculating a Mandel net residualerror (D_(M)) and a reference measurement system uncertainty (σ_(RMS))for each measurement; calculating a variance for each D_(M) and σ_(RMS);constructing a Chi-squared probability distribution function (χ²-pdf)for the D_(M) variances and the σ_(RMS) variances; calculating a χ²-pdffor the TMU according to the equationpdf _(TMU)=χ_(D) _(M) ₂ ²{circle over (×)}χ_(σ) _(RMS) ₂ ², whereinpdf_(TMU) is the TMU χ²-pdf, χ_(D) _(M) ₂ ² is the D_(M) varianceχ²-pdf, χ_(σ) _(RMS) ₂ ² is the σ_(RMS) variance χ²-pdf, and {circleover (×)} is a convolution operation; and determining a TMU range frompdf_(TMU), wherein the TMU range determines the uncertainty in theestimated TMU.
 26. The method of claim 25, wherein the arms is definedas one of: an RMS precision; an independently determined RMS totalmeasurement uncertainty (TMU_(RMS)); and V_(U) _(RMS) =V_(ST)+V_(AG),wherein V_(U) _(RMS) is U_(RMS) expressed as a variance, V_(ST) is ashort term precision variance, and V_(AG) is an across grating variance.27. The method of claim 26, wherein V_(U) _(RMS) further includes atleast one of a multiple reference system variance (V_(MRMS)) and a longterm precision variance V_(LT) according to the equationV _(U) _(RMS) =V _(ST) +V _(AG) +V _(MRMS) +V _(LT).
 28. The method ofclaim 25, wherein the plurality of measurements are of structures thatrepresent variations in a semiconductor process.
 29. A method fordetermining an uncertainty in a total measurement uncertainty (TMU), themethod comprising the steps of: determining a confidence limit for a netresidual error from a Mandel analysis (D_(M)) using chi-squareddistribution tables; and determining a confidence limit for a totalmeasurement uncertainty (σ_(TMU)) according to the equationσ_(TMU) ={square root}{square root over (D _(M) ² −U _(RMS) ² )}, whereU_(RMS) is a measurement uncertainty of a reference measurement system.30. The method of claim 29, wherein each of D_(M) and U_(RMS) isdetermined by at least about 40 measurements and wherein σ_(TMU) isrepresented as σ_(TMU)±δ_(TMU); U_(RMS) is represented asU_(RMS)±δ_(RMS); D_(M) is represented as D_(M)±δ_(Mandel); and theequation σ_(TMU)={square root}{square root over (D_(M) ²−U_(RMS) ²)} isrepresented as$\delta_{TMU} = {\sqrt{{\frac{D_{M}^{2}}{\sigma_{TMU}^{2}}\delta_{Mandel}^{2}} + {\frac{U_{R\quad{MS}}^{2}}{\sigma_{TMU}^{2}}\delta_{R\quad{MS}}^{2}}}.}$